A Local-global Principle for Diophantine Equations

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Abstract

We observe the following global-local principle for Diophantine
equations: If an equation f(x₁,...,x[subscript t]) ϵ ℤ[x₁,...,x[subscript t] = p can be
solved efficiently for a dense set of primes p, then one can efficiently
obtain solutions to equations of the form
f(x₁,...,x[subscript t]) Ξ r mod n.
This is done without any knowledge of the factorization of n.
We apply this principle to get the following results:
- There is an efficient algorithm to solve a modular version of Fermat's
equation
x² + y² Ξ r (mod n).
- Assuming factoring is hard, it is hard to solve the following equation
for a dense set of primes p :
x² - y² = p - 1.
Randomness and Extended Riemann Hypothesis are essential in the
proofs.